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Theorem dmdi2 29714
 Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))

Proof of Theorem dmdi2
StepHypRef Expression
1 dmdi 29712 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
2 eqimss2 3883 . 2 (((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
31, 2syl 17 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ∩ cin 3797   ⊆ wss 3798   class class class wbr 4875  (class class class)co 6910   Cℋ cch 28337   ∨ℋ chj 28341   𝑀ℋ* cdmd 28375 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-iota 6090  df-fv 6135  df-ov 6913  df-dmd 29691 This theorem is referenced by: (None)
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